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\author{学号 \underline{\hspace{4cm}} \hspace{1cm} 姓名 \underline{\hspace{4cm}} }
\title{复变函数第二章部分习题}
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\date{2024 年 3 月 25 日}
%\date{March 9, 2021}

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\begin{document}

\maketitle

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\begin{enumerate}

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\item[3.]  %Problem 01
设函数 $f(z)$ 如下，证明 $f(z)$ 在原点满足柯西-黎曼方程，但是不可微。
$$f(z) = \left\{ \begin{array}{ll}
\frac{x^3-y^3+i(x^3+y^3)}{x^2+y^2}, & z=x+iy\neq 0, \\
0, & z=0,
\end{array}\right. $$


\vspace{0.2cm}

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\item[4.]  %Problem 02
证明下述函数在 $z$ 平面上任何点都不解析：
$
(1) |z|;\,\, 
(2) x+y; \,\, 
(3) \mathrm{Re}(z); \,\, 
(4) \frac{1}{\bar{z}}. 
$

\vspace{0.2cm}

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\item[5.]  %Problem 03
判断下列函数的可微性与解析性：
%\begin{eqnarray*}
\begin{enumerate}[label={(\arabic*)}]
\item  $f(z)=xy^2+ix^2y$. 
\item  $f(z)=x^2+iy^2$. 
\item  $f(z)=2x^3+3iy^3$. 
\item  $f(z)=x^3-3xy^2+i(3x^2y-y^3)$. 
\end{enumerate} 
%\end{eqnarray*}

\vspace{0.2cm}

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\item[8.]  %Problem 04
证明函数 $f(z) = x^3+3x^2yi-3xy^2-y^3i$ 在复平面上解析，并求其导数。

\vspace{0.2cm}

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\item[10.]  %Problem 05
设 $z=x+iy$, 求 
$
(1)\,\, |\exp(i-2z)|; \,\,
(2)\,\, |\exp(z^2)|; \,\,
(3)\,\, \mathrm{Re}(\exp(1/z)). 
$

\vspace{0.2cm}

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\item[11.]  %Problem 06
求下列值及其主值：
$
(1)\,\, \mathrm{Ln}\, (3-\sqrt{3}i) ; \,\,
(2)\,\, (2i)^i. 
$

\vspace{0.2cm}

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\item[14.]  %Problem 07
证明：
$
(1)\,\, \lim\limits_{z\to 0} \frac{\sin z}{z}=1; \hspace{0.3cm}
(2)\,\, \lim\limits_{z\to 0} \frac{\exp(z)-1}{z}=1; \hspace{0.3cm}
(3)\,\, \lim\limits_{z\to 0} \frac{z-z\cos z}{z-\sin z}=3. 
$

\vspace{0.2cm}

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\item[20.]  %Problem 08
解方程：
%\begin{eqnarray*}
\begin{enumerate}[label={(\arabic*)}]
\item  $\exp(z)=1+\sqrt{3}i$. 
\item  $\ln z = \pi i/2$. 
\item  $1+\exp(z)=0$. 
\item  $\cos(z)+\sin(z)=0$. 
\item  $\tan(z)=1+2i$. 
\end{enumerate} 
%\end{eqnarray*}

\vspace{0.2cm}

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\item[21.]  %Problem 09
设 $z=re^{i\theta}$, 证明 $\mathrm{Re}[\ln(z-1)] = \frac{1}{2} \ln (1+r^2-2r\cos\theta). $

\vspace{0.2cm}

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\item[22.]  %Problem 10
设 $w=\sqrt[3]{z}$ 确定在 $\mathbb{C}-[0,\infty)$ 上，并且 $w(i)=-i$, 求 $w(-i)$.

\vspace{0.2cm}


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\item[24.]  %Problem 10
求 $(1+i)^i$ 与 $3^i$ 的值。

\vspace{0.2cm}


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\item[H2.]  %Problem 10
设 $f(z)=\frac{z}{1-z}$, 证明当 $|z|<1$ 时，有
$\mathrm{Re} \left[ 1+ z\frac{f\,''(z)}{f\,'(z)} \right] >0. $

\vspace{0.2cm}


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\item[H4.]  %Problem 10
设 $f(z)=u+iv\in C^1(D)$, 证明： 
$$ \begin{vmatrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} \\ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} \end{vmatrix}  = \left\lvert \frac{\partial f}{\partial z}\right\rvert^2 
-\left\lvert \frac{\partial f}{\partial \bar{z}}\right\rvert^2. $$ 

\vspace{0.2cm}


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\end{enumerate}


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\end{document}

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